使用中央差分法計算長時間的不可壓縮流場演變時，為免產生數值不穩定，需滿足離散動能守恆。如果算則能同時滿足離散質量、動量及動能守恆，則稱為完全守恆則。
在本文中，吾人將適用於均勻格點的完全守恆算則推廣至非均勻格點。其中對流項需要特別的處理，經由對交錯式格點系統上的 Harlow and Welsh 二階算則的分析，發現必須將通量速度表示為質量通量，才能維持離散動能恆。

要將 Morinishi 等人的高階對流項算則，推廣至非均勻交錯式格點系統，需要兩項修改，第一項為必須在計算空間操作以保存離散運算子的對稱性，第二項為對流項中的速度形式必須表示為逆變速度-卡氏速度的組合，因為逆變速度和質量通量有相同的結構。相同的方法可用於純量變數的對流項。吾人使用不同的數值測試去驗證新算則的守恆性、正確性及效能，及使用大渦法模擬管道流，以了解格點及次格點尺度運動對紊流統計的影響。

本文另一目的是使用使用大渦法模擬垂直管道內之完全發展紊混合對流，其中 Re_b = 5600, Pr = 0.71 且兩壁的熱通量為定值。

對於順向熱流場而言，存在一轉變 Gr_q 數，其值約為 1.40x10^8。當 Gr_q 比轉變 Gr_q 數小時，熱流場的結構近似純強制對流，但強度減弱，Nu 數變小。 當 Gr_q 為轉變 Gr_q 數時，近牆的紊流結構的再產生過程，大部份遭破壞，使得近牆處二階統計量大幅下降，Nu 數也只有純強制對流 45%。在遠離壁面處，浮力的效應明顯影響紊流統計。最大平均速度開始遠離中心，導致雷諾剪應力及流線方向紊流熱通量變為負值，浮力生產項變為正值。u' 和 theta' 的相似性開始退化。當 Gr_q 比轉變 Gr_q 數大時，浮力對紊流統計量的影響逐漸增強，Nu 數也隨 Gr_q 數的增加而增加，u' 和 theta' 的相似性也逐漸降低。

對於逆向熱流場而言，浮力的方向和雷諾剪應力相反，使得浮力的增加，會造成紊流強度的增加，且紊流統計的分佈相似於純強制對流，Nu 數也隨 Gr_q 數的增加而增加。近牆的低速煙線結構相似於純強制對流，但長度變短，在最高計算的 Gr_q 數，u' 和 theta' 的相似性開始退化。

A central difference (dissipation-free) scheme has to conserve kinetic energy to avoid numerical instability when a long-term time integration for incompressible flow such as large eddy simulation or direct numerical simulation is performed. A scheme is called fully conservative if it can simultaneously conserve mass, momentum, and kinetic energy in the discrete sense. A theoretical analysis is performed to extend the fully conservative schemes to non-uniform grid systems without sacrificing any conservative properties. The main step is to design a convective scheme which conserves momentum and kinetic energy simultaneously. An analysis is made for the second-order accurate scheme of Harlow and Welsh for staggered grid systems and it is found that the flux velocities need to be viewed as mass fluxes across control surfaces to conserve kinetic energy.

To extend the analysis to higher order schemes, it is necessary to work in computational space. The contravariant-Cartensian velocity formulation for the convection term in computational space has the similar structure for the proposed fully conservative second-order scheme in physical space. Using the velocity formulation, the higher order convective schemes of Morinishi are extended to non-uniform staggered grid systems for the advective, divergence and skew-symmetric forms. The higher order schemes for scalar variables which conserve the square of the scalar variables are also derived. Several numerical tests are used to validate the conservative properties, accuracy and performance of the proposed higher order schemes. A series of LESs of turbulent heat transfer in channel flow to study the contributions of SGS motions and the influences of grid number on turbulent statistics.

Large eddy simulations are performed to study fully developed turbulent mixed convection in a vertical plane channel, Re_b = 5600 and Pr = 0.71, with uniform heating or cooling from both walls. The main features of turbulent mixed convection are produced.

For aiding flow, a transition Gr_q number, Gr_q = 1.40x10^8, exists. Before the transition number, the turbulence is generated mostly by the shear force driven by the pressure gradient. The turbulent statistics are similar in shape to those for forced convection while the magnitudes reduce slightly in the near-wall region for all turbulent statistics and the friction coefficient and the Nusselt number also decrease gradually. The buoyancy production term in the budget of turbulent kinetic energy remain small and negative over the whole channel.

Around the transition Gr_q number, the regeneration process of near-wall structures are destroyed mostly. Second order statistics show the severest deterioration in the near-wall region and the turbulence generated by buoyancy becomes apparent on turbulent statistics away from the wall. The friction coefficient and the Nusselt number decline to 85% and 45%, respectively, of that at Gr_q=0. The point of
the maximum mean velocity begins to shift away from the channel center and the Reynolds shear stress and streamwise turbulent heat flux change sign nearly at the location of the maximum mean velocity. The buoyancy production term changes sign, and thus the term becomes a main producing term while y is larger than the zero point. The similarity between u' and theta' begins to deteriorate.

After the transition Gr_q number, turbulence generated by buoyancy gradually increases its influence on turbulent statistics. The magnitudes increase gradually in the near-wall region for all turbulent statistics, the friction coefficient and the Nusselt number with increasing Gr_q number. The dissimilarity between u' and theta' increases gradually and the thermal plumes become the main structures at highest simulated Gr_q.

For opposing flow, the contributions of the buoyant force and Reynolds shear stress are in the opposite direction, and thus the turbulence intensity increases as the buoyant force increases. The turbulent statistics are similar in shape to those for forced convection while the magnitudes increase in the near-wall region for all turbulent statistics except for the mean streamwise velocity and the friction coefficient, and the Nusselt number also increases gradually with increasing Gr_q number. The near-wall streaky structures are similar to those of Gr_q = 0, but the dissimilarity between u' and theta' is observed at the highest simulated Gr_q.

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